Coupling and perturbation techniques for categorical time series (Q2203638)

This paper deals with categorical time series, a particular topic in stochastic processes but of wide application in several scientific fields, among others in meteorology. The author proposes the following model for categorical time series coupled to an exogenous covariate process. Let $(\Omega,F,P)$ be a probability space, $\mathbb{N}=\{0,1,2,\dots\}$, $E$ a finite set, for each $t\in\mathbb{Z}$ let $X_{t}:\Omega\to\mathbb{R}^{d}$ be a random vector, $Y_{t}:\Omega\to E$ a random field with values in $E$ (over $E$ is considered the $\sigma$-algebra of all the subsets of $E$), $X=(X_{t})_{t\in\mathbb{Z}}$, $Y=(Y_{t})_{t\in\mathbb{Z}}$, $D\in(B(\mathbb{R}^{d}))^{\mathbb{Z}}$ such that $P(X\in D)=1$, $q:E\times E^{\mathbb{N}}\times D\to(0,1)$ a measurable function with \[ \sum _{e\in E}q(e|y,x)=1\quad (y,x)\in E^{\mathbb{N}}\times D. \] If $t\in\mathbb{Z}$ and $e\in E$, then \[ P(Y_{t}=e|Y_{t-1}^-,X)=q(e|Y_{t-1}^-,X_{t-1}^-). \tag{1} \] where \[ Y_{t-1}^-=(Y_{s})_{s\leq t-1},\quad X_{t-1}^-=(X_{s})_{s\leq t-1}. \] Under the following assumptions it is shown that there exists and is unique a process $Y$ satisfying (1) S1. $X$ is a stationary and ergodic process. S2. Let $b_{m}$ defined as \[ b_{m}:=\sup\{(1/2)\sum_{e\in E}|q(e|y,x_{t}^-)-q(e|y',x_{t}^-)|\}, \] $(y,y',x)\in E^{\mathbb{N}}\times E^{\mathbb{N}}\times D$; $t\in\mathbb{Z}$; $y_{i}=y_{i}'$ where $1\leq i\leq m$ for each $m\in\mathbb{N}$ with $b_0<1$ and $\sum_{m\geq 0}b_{m}<\infty$. That model is shown in Section 4 to be well suited to series of practical interest in various disciplines. A model similar to this one has been used in econometrics to analyze changes in product prices and to predict recessions. Section 5 is concerned with the study of some dependence properties between $Y$ and $X$ processes that are useful for analyzing their asymptotic behavior. Some possible applications in statistical inference of the models and results presented are given in the following section. The last section presents the detailed and refined proofs of the most important results of the paper. Finally, the list of references is extensive and updated.